PDE-OBS: Controlled Evaluation Across Observation Patterns

Ruichen Xu; Siyao Wang; Fang Wan; Jiacheng Qiu; Wenhan Gao; Jiaxing Zhang; Linsey Pang; Ravid Shwartz-Ziv

cs.LGSep 29, 2026

PDE-OBS: Controlled Evaluation Across Observation Patterns

Authors: Ruichen Xu, Siyao Wang, Fang Wan, Jiacheng Qiu, Wenhan Gao, Jiaxing Zhang, Linsey Pang, Ravid Shwartz-Ziv, +2 more

Organizations: Stony Brook University · University of California, Davis · Independent Research · PayPal · Northeastern University · New York University

Abstract

Physical-field reconstruction and forecasting depend on both measurement density and spatial layout, yet evaluation under a single observation pattern does not characterize performance when that pattern changes. We introduce PDE-OBS, an integrated benchmarking platform spanning numerical data generation, model training, and inference and evaluation under varying observation conditions. It combines 560,000 fields and trajectories from seven partial differential equation families with configurable observation operators and seven adapted baseline methods for stationary reconstruction and short-horizon forecasting. Separating observation construction from physical records allows users to specify parameterized patterns and deterministic mixtures for training and testing while preserving prediction targets and data splits. The evaluation protocol uses references trained for each test pattern to compare models on identical test observations and targets, alongside equal-count groups for spatial-layout comparisons. On a 14,000-record subset, we evaluate 441 trained models under nine test patterns, yielding 3,969 evaluations. Mean cross-pattern error exceeds mean matched-pattern error in all 49 PDE-method pairs, and this finding persists in a configuration-matched subset of 117 models. Denser test observations do not consistently reduce error for a fixed model. Mixed-pattern training on five completed pairs reduces large single-pattern transfer errors, although destination-trained references usually remain more accurate. Together, the benchmark and findings support systematic evaluation of observation-pattern sensitivity and provide a reusable workflow for developing methods under changing measurement conditions. Code: https://github.com/ru1ch3n/PDE-OBS.

Figures & tables

Figure 1: Overview of PDE-OBS . The benchmark connects numerical data generation, observation construction, model training, and inference across observation conditions. (a) Numerical solvers generate complete fields and trajectories with fixed train/test splits. (b) Different observation patterns are applied to the same physical record while prediction targets remain unchanged. The observed field is the solution for reconstruction and the initial state for forecasting. (c) Models perform stationary reconstruction or autoregressive forecasting and are evaluated under matched and shifted observation patterns. For each test pattern, a reference model is trained on the training split using that pattern and evaluated on identical test observations and targets. Field maps are schematic; Section 3 specifies the benchmark tasks and observation protocols.

Figure 2: Performance across training and test observation patterns. Each PDE–method panel contains a $9\times 9$ matrix with training patterns in rows and test patterns in columns, both ordered R50, H, V, CL, BL, LI, BD, R65, R80. Color denotes mean relative- $L^{2}$ error over 200 test records on a shared absolute logarithmic scale; darker colors indicate larger errors. Outlined cells ( $\square$ ) identify matched training and test patterns. Dots ( $\bullet$ ) identify the 13 PDE–method pairs in the configuration-matched subset. Thin separators distinguish the two equal-count layout groups and the denser random patterns. The figure contains all 3,969 evaluations. Per-cell means, sample standard deviations, and training configurations are reported in Appendix F . CNO denotes the CNO-inspired adaptation.

Figure 3: Single-pattern and mixed-pattern training: five completed pairs. Top: nine single-pattern rows and one additional MIX row, with test patterns in columns and the shared error scale of Figure 2 . Outlines mark matched cells. Bottom: MIX ratios to the destination-trained reference (dots) versus the range and median of eight transferred models (bars); the dashed line is $R=1$ . Poisson’s displayed geometric mean uses all nine destinations descriptively; its BL/BD/CL recipe-matched mean is 3.05. MIX rows supplement, rather than replace, the 441-model grid (Appendix G ).

Appendix figures & tables67 assets

Supplementary material from the paper’s appendix.

Appendix

Family	Prediction	Boundary	$T$
Darcy	Solution-field recovery	Dirichlet	–
Poisson	Solution-field recovery	Dirichlet	–
Helmholtz	Solution-field recovery	Dirichlet	–
Heat	Initial field to three future states	Periodic	.25
Reaction–diffusion	Initial field to three future states	Periodic	.8
Burgers	Initial field to three future states	Periodic	.35

Appendix

Table 1: Evaluated task definitions. Each family uses 2,000 scalar fields or trajectories on a

128\times 128

grid. Temporal targets are frames 1–3 of 15 equally spaced frames,

t_{h}=hT/14

. All inputs are noise-free partial observations.

Family	Parameter	Values
Darcy	coefficient contrast $\chi$	2, 8, 32
Poisson	source amplitude	0.5, 1, 2
Helmholtz	wavenumber $k$ ; damping	0.5, 1, 2; zero
Heat	diffusivity $\alpha$	0.005, 0.02, 0.08
Reaction–diffusion	rate $\rho$ ; diffusivity	0.5, 1.5, 4; 0.008
Burgers	viscosity $\nu$	0.04, 0.015, 0.004

Appendix

Table 2: Parameters in low/medium/high order. Regime names are family-specific, not a common difficulty scale.

PDE	Mean normalized residual	Maximum normalized residual
Darcy	$1.14\times 10^{-5}$	$1.57\times 10^{-5}$
Poisson	$1.69\times 10^{-5}$	$2.84\times 10^{-5}$
Helmholtz	$1.80\times 10^{-5}$	$3.52\times 10^{-5}$
Heat	$1.32\times 10^{-8}$	$1.51\times 10^{-8}$
Reaction–diffusion	$3.39\times 10^{-5}$	$1.55\times 10^{-4}$
Burgers	$1.26\times 10^{-2}$	$3.85\times 10^{-2}$

Appendix

Table 3: Recorded production consistency for the evaluated slice (2,000 records per family). Residuals use the family-specific discrete operator and normalization, not a common continuum-error scale.

PDE	Final $T$	Dense frames	Stride	Stored frames	Internal rule
Heat	0.25	155	11	15	Fourier propagation
Reaction–diffusion	0.8	267	19	15	$\rho\delta\leq 0.18$
Burgers	0.35	267	19	15	CFL bound above
Navier–Stokes	0.4	71	5	15	$\delta\leq 10^{-4}$

Appendix

Table 4: Time grids in the evaluated periodic slice. Dense quality frames include both endpoints; stored data select every indicated stride. Internal substeps are separate from both grids.

Figure 4: Mean recorded PDE residuals. Each bar is the arithmetic mean over 2,000 selected records, shown on a logarithmic axis. Six family means are below

1\%

; Burgers is

1.26\%

. The dashed line marks the

1\%

reference and the dotted line the original

5\%

generation gate. Residual operators and normalizations are family-specific, with a semigroup defect for Heat; these values measure production discrete consistency, not independent solution error.

Code	Pattern	Count $K$	Construction
R50	Random	8192	Uniform distinct locations
R65	Random	10650	Uniform distinct locations
R80	Random	13107	Uniform distinct locations
BL	Missing block	8284	Outside a missing $90\times 90$ square
LI	Crossed lines	8284	Union of 38 rows and 38 columns
H	Horizontal lines	8192	64 distinct rows

Appendix

Table 5: Exact observation counts out of 16,384 locations. Equal-count comparisons use R50/H/V/CL or BL/LI/BD, not all nominal 50% masks together. A perimeter band on a periodic grid is not a physical wall.

Adaptation	Reference configuration	Parameters	Batch
U-FNO-2D	Width 36; Fourier and U blocks	5,049,545	4
FNO	Width 32; 12 modes; 4 layers	2,366,145	8
CNO-inspired CNN	Width 32; filtered convolution	103,617	8
DeepONet	Spatial CNN branch; latent 128	221,121	4
GNOT	Width 128; 3 layers; 2 experts	2,099,975	4
Transolver (static)	Width 128; 8 layers; 64 slices	2,811,457	4

Appendix

Table 6: Frozen reference model sizes and nominal per-device batch sizes. These describe the reference registry, not a proof of identical effective batches or capacities in every execution attempt. Complex weights are counted as real scalars. Attempt-specific configurations remain necessary for reproduction.

Method	Optimizer	Weight decay	Batch	Schedule	Grad. clip
U-FNO-2D	Adam	$10^{-4}$	4	step	1.0
FNO	Adam	$10^{-4}$	8	step	1.0
CNO-inspired CNN	AdamW	$10^{-6}$	8	step	1.0
DeepONet	Adam	0	4	step	1.0
GNOT	AdamW	$5\times 10^{-5}$	4	one-cycle, 20% warm-up	1000
Transolver	AdamW	$10^{-5}$	4 (2)	one-cycle, 30% warm-up	0.1 (none)

Appendix

Table 7: Original optimization defaults. Step schedules halve the rate every 100 epochs; one-cycle schedules advance per batch with cosine annealing. Parentheses denote temporal Transolver settings.

Factory	Controls
Random	ratio or count ; distinct cells
Grid	ratio or spacing ; optional random_offset
Block	missing_fraction or block_shape ; observe outside the block
Line / horizontal / vertical	ratio or num_lines ; orientation for general lines
Boundary	width ; optional deterministic count within the band
Clustered	ratio or count ; clusters and spread

Appendix

Table 8: General observation controls. Record realized locations and counts; a requested ratio need not equal realized density. Alternatives separated by or are mutually exclusive.

Statistic	Full grid	Fixed subset
Pairs	49	13
Pairs with $C>D$	49/49	13/13
Median $C/D$	10.46	10.74
Positive layout directions, group I	545/588	151/156
Positive layout directions, group II	276/294	76/78
R80 improves over R50, R50-trained	13/49	0/13

Appendix

Table 9: Sensitivity to fixed original500 training provenance. Counts and medians summarize dependent comparisons, not independent experimental replications. The metadata-defined subset is unchanged.

	Relative- $L^{2}$ test error		Transfer ratio
PDE	Matched	Cross-pattern	Median $C/D$
Stationary solution recovery
Darcy	$0.08\pm 0.10$	$0.80\pm 2.40$	13.45
Poisson	$0.06\pm 0.07$	$1.50\pm 5.71$	16.58
Helmholtz	$0.05\pm 0.14$	$4.91\pm 30.45$	28.61
Temporal state forecasting

Appendix

Table 10: Absolute error and transfer sensitivity from new predictions. Matched/cross-pattern entries are mean

\pm

sample SD over 63/504 cell means per PDE (seven methods); the ratio is the median of seven pairwise

C/D

values. These are not error percentages, seed uncertainty, or an architecture ranking.

Code	Recorded cohort	Models
O500	original500	257
INT	interrupted_or_recovered	28
M120	budget200_min120	44
P10	budget200_patience10_no_floor	10
R500	original500_recovered	66
F200	fixed200_recovered	36

Appendix

Table 11: Retained training-provenance groups. Codes identify histories, not scientific quality; every retained model contributes all nine test views.

Equal-count layout: positive destination-referenced difference
Condition	Comparison	Count / total	Median ratio
$K=8,192$	Recovery	238/252	7.53
$K=8,192$	Rollout	307/336	2.97
$K=8,284$	Recovery	122/126	12.37
$K=8,284$	Rollout	154/168	3.76
Density: lower error than the same checkpoint at R50

Appendix

Table 12: Layout and density contrasts. The first group counts

E_{v,w}>E_{w,w}

and summarizes

E_{v,w}/E_{w,w}

; the second counts

E_{v,w}<E_{v,\mathrm{R50}}

and summarizes

E_{v,w}/E_{v,\mathrm{R50}}

. Density masks are sampled separately and are not constructed as nested sets; statistical independence is not assumed. Counts are integers; ratios are dimensionless.

	Mean $\pm$ SD			Cross-pattern cell means
Method	Matched $D$	Cross $C$	$C/D$	Median	90th pct.	Max.
Darcy
U-FNO	$0.26\pm 0.10$	$0.51\pm 0.32$	2.01	0.36	1.07	1.49
FNO	$0.08\pm 0.11$	$0.81\pm 0.86$	10.46	0.54	1.80	4.39
CNO	$0.04\pm 0.05$	$0.48\pm 0.53$	10.74	0.37	0.82	3.63
DeepONet	$0.06\pm 0.04$	$1.58\pm 2.95$	26.63	0.44	3.67	14.26

Appendix

Table 13: All 49 PDE–method summaries. Relative-

L^{2}

errors are displayed to two decimals.

D

C

are mean

\pm

sample SD across 9/72 cell means; the last three columns describe the 72 cross-pattern means. Blue retains the 13 fixed original500 pairs, not a winner or a uniform-budget architecture ranking. All medians, ratios, and tails are calculated before rounding.

	Horizon 1		Horizon 2		Horizon 3
Method	$D_{1}$	$C_{1}$	$D_{2}$	$C_{2}$	$D_{3}$	$C_{3}$
Heat
U-FNO	0.16±0.12	0.33±0.24	0.16±0.12	0.32±0.23	0.18±0.11	0.32±0.22
FNO	0.03±0.01	0.19±0.21	0.05±0.01	0.19±0.21	0.08±0.01	0.20±0.20
CNO	0.05±0.03	0.26±0.25	0.06±0.02	0.26±0.23	0.09±0.02	0.27±0.22
DeepONet	0.08±0.04	1.40±1.97	0.09±0.04	2.89±6.12	0.12±0.04	10.66±29.76

Appendix

Table 14: All forecasting horizons in one grouped table. Each entry reports mean

\pm

sample SD of 9 matched or 72 cross-pattern cell means on one line, with two decimals in both terms. Gray retains the method grouping. These relative-

L^{2}

errors are auxiliary per-frame scores, not components to average into the joint rollout score.

	Test observation pattern
Train	R50	H	V	CL	BL	LI	BD	R65	R80
R50	0.18±0.03	0.25±0.06	0.24±0.06	0.76±0.15	0.76±0.07	0.24±0.05	0.73±0.06	0.31±0.02	0.57±0.04
H	0.32±0.05	0.32±0.05	0.38±0.07	1.08±0.30	1.49±0.30	0.35±0.07	1.47±0.26	0.31±0.06	0.32±0.05
V	0.36±0.18	0.36±0.18	0.36±0.18	0.36±0.18	0.36±0.18	0.36±0.18	0.36±0.18	0.36±0.18	0.36±0.18
CL	0.27±0.14	0.27±0.14	0.27±0.14	0.30±0.14	0.30±0.15	0.27±0.14	0.31±0.16	0.29±0.12	0.34±0.10
BL	0.36±0.18	0.36±0.18	0.36±0.18	0.36±0.18	0.36±0.18	0.36±0.18	0.36±0.18	0.36±0.18	0.36±0.18

Appendix

Table 15: Darcy / U-FNO. All 81 cells: mean

\pm

SD on one line (

n=200

); blue marks matched views. All training views: 500/O500.

	Test observation pattern
Train	R50	H	V	CL	BL	LI	BD	R65	R80
R50	0.02±0.01	0.16±0.07	0.20±0.09	1.87±0.50	1.81±0.33	0.14±0.05	1.73±0.29	0.39±0.04	1.23±0.16
H	0.15±0.07	0.03±0.02	1.44±0.81	2.98±0.98	4.28±0.80	0.99±0.61	4.39±0.81	0.34±0.06	0.48±0.09
V	0.07±0.02	0.48±0.23	0.02±0.01	2.52±0.61	2.17±0.41	0.36±0.19	2.21±0.40	0.88±0.12	0.95±0.16
CL	0.20±0.05	0.20±0.06	0.19±0.06	0.14±0.09	0.26±0.10	0.20±0.06	0.35±0.07	0.26±0.03	0.35±0.05
BL	0.48±0.12	1.05±0.45	0.80±0.30	0.60±0.16	0.08±0.04	0.76±0.31	0.07±0.04	0.42±0.12	0.37±0.09

Appendix

Table 16: Darcy / FNO. All 81 cells: mean

\pm

SD on one line (

n=200

); blue marks matched views. All training views: 500/O500.

	Test observation pattern
Train	R50	H	V	CL	BL	LI	BD	R65	R80
R50	0.01±0.00	0.10±0.05	0.09±0.04	0.56±0.15	0.73±0.11	0.08±0.04	0.80±0.06	1.77±0.76	3.63±0.45
H	0.06±0.01	0.01±0.01	0.19±0.04	0.38±0.19	0.55±0.13	0.12±0.03	0.59±0.07	0.93±0.09	0.96±0.13
V	0.02±0.00	0.08±0.03	0.02±0.01	0.36±0.15	0.48±0.10	0.05±0.01	0.51±0.07	0.05±0.01	0.08±0.01
CL	0.07±0.01	0.07±0.02	0.08±0.02	0.11±0.09	0.29±0.16	0.07±0.02	0.36±0.15	0.08±0.01	0.10±0.02
BL	0.48±0.11	0.38±0.08	0.37±0.08	0.26±0.10	0.15±0.09	0.39±0.08	0.14±0.09	0.57±0.14	0.63±0.16

Appendix

Table 17: Darcy / CNO. All 81 cells: mean

\pm

SD on one line (

n=200

); blue marks matched views. All training views: 500/O500.

	Test observation pattern
Train	R50	H	V	CL	BL	LI	BD	R65	R80
R50	0.04±0.02	0.14±0.04	0.16±0.04	0.58±0.11	0.71±0.08	0.11±0.03	0.74±0.04	0.14±0.01	0.20±0.02
H	0.26±0.06	0.04±0.02	0.69±0.24	0.80±0.26	0.80±0.17	0.36±0.13	0.69±0.12	0.34±0.07	0.40±0.07
V	0.33±0.02	0.80±0.32	0.05±0.02	0.48±0.15	0.44±0.14	0.45±0.20	0.32±0.10	0.18±0.02	0.15±0.03
CL	0.24±0.06	2.88±1.08	1.59±0.44	0.14±0.07	0.29±0.10	0.58±0.19	0.31±0.07	0.22±0.06	0.23±0.06
BL	8.56±1.10	12.22±2.47	14.26±2.39	1.87±0.77	0.10±0.06	13.56±2.00	0.08±0.04	7.52±0.96	4.92±0.68

Appendix

Table 18: Darcy / DeepONet. All 81 cells: mean

\pm

SD on one line (

n=200

); blue marks matched views. Epoch/cohort in row order: 500/O500, 500/O500, 290/INT, 500/O500, 500/O500, 500/O500, 500/O500, 500/O500, 500/O500.

	Test observation pattern
Train	R50	H	V	CL	BL	LI	BD	R65	R80
R50	0.01±0.01	0.03±0.01	0.03±0.01	0.54±1.57	2.47±7.11	0.03±0.01	0.33±0.24	0.01±0.01	0.01±0.01
H	0.08±0.01	0.08±0.01	0.10±0.02	0.31±0.20	0.35±0.12	0.09±0.02	0.35±0.09	0.08±0.01	0.08±0.01
V	0.04±0.01	0.06±0.02	0.05±0.01	0.26±0.18	0.33±0.12	0.05±0.02	0.34±0.07	0.04±0.01	0.04±0.01
CL	0.06±0.03	0.06±0.03	0.06±0.03	0.10±0.07	0.23±0.10	0.06±0.03	0.24±0.07	0.06±0.03	0.06±0.03
BL	0.27±0.07	0.28±0.07	0.27±0.07	0.41±0.17	0.11±0.08	0.28±0.07	0.08±0.07	0.27±0.07	0.27±0.07

Appendix

Table 19: Darcy / GNOT. All 81 cells: mean

\pm

SD on one line (

n=200

); blue marks matched views. Epoch/cohort in row order: 500/O500, 185/INT, 182/INT, 184/INT, 500/O500, 500/O500, 185/INT, 500/O500, 500/O500.

	Test observation pattern
Train	R50	H	V	CL	BL	LI	BD	R65	R80
R50	0.00±0.00	0.52±0.16	0.60±0.20	1.31±0.36	1.53±0.31	0.53±0.11	1.55±0.26	0.05±0.01	0.13±0.02
H	0.19±0.03	0.00±0.01	0.29±0.04	0.33±0.13	0.43±0.10	0.10±0.02	0.46±0.08	0.17±0.03	0.15±0.03
V	0.40±0.04	0.46±0.05	0.01±0.00	0.64±0.19	0.73±0.15	0.31±0.04	0.80±0.10	0.34±0.04	0.27±0.03
CL	0.02±0.00	0.04±0.02	0.04±0.02	0.06±0.06	0.19±0.10	0.04±0.01	0.18±0.09	0.04±0.01	0.06±0.01
BL	0.69±0.17	0.82±0.21	0.66±0.18	0.54±0.24	0.08±0.05	0.57±0.16	0.06±0.04	0.61±0.15	0.55±0.13

Appendix

Table 20: Darcy / Transolver. All 81 cells: mean

\pm

SD on one line (

n=200

); blue marks matched views. Epoch/cohort in row order: 500/O500, 500/O500, 500/O500, 500/O500, 500/O500, 500/O500, 200/M120, 500/O500, 500/O500.

	Test observation pattern
Train	R50	H	V	CL	BL	LI	BD	R65	R80
R50	0.02±0.01	0.08±0.06	0.08±0.05	0.92±0.36	1.34±0.24	0.07±0.05	1.50±0.22	0.29±0.01	0.57±0.03
H	0.10±0.02	0.02±0.01	0.24±0.07	1.05±0.28	1.09±0.15	0.17±0.06	1.10±0.08	0.29±0.02	0.53±0.04
V	0.50±0.12	0.25±0.08	0.02±0.01	37.80±27.49	2.20±1.19	0.27±0.11	2.73±0.67	0.53±0.11	0.63±0.09
CL	0.18±0.08	0.15±0.07	0.15±0.07	0.14±0.09	0.29±0.10	0.15±0.07	0.34±0.08	0.13±0.05	0.16±0.05
BL	0.38±0.07	0.30±0.09	0.31±0.10	0.40±0.12	0.04±0.04	0.29±0.08	0.03±0.03	0.31±0.07	0.24±0.06

Appendix

Table 21: Darcy / PINO. All 81 cells: mean

\pm

SD on one line (

n=200

); blue marks matched views. All training views: 500/O500.

	Test observation pattern
Train	R50	H	V	CL	BL	LI	BD	R65	R80
R50	0.18±0.11	0.20±0.13	0.21±0.13	2.69±1.76	3.28±2.08	0.19±0.11	3.11±1.98	0.12±0.06	1.00±0.64
H	0.50±0.14	0.08±0.05	0.57±0.11	0.49±0.15	0.73±0.21	0.24±0.06	0.77±0.20	0.46±0.14	0.42±0.12
V	0.62±0.24	0.61±0.24	0.06±0.03	0.74±0.28	0.90±0.36	0.35±0.13	0.93±0.35	0.60±0.24	0.62±0.24
CL	0.06±0.03	0.07±0.02	0.07±0.03	0.05±0.04	0.38±0.15	0.05±0.02	0.39±0.14	0.05±0.02	0.04±0.01
BL	0.31±0.05	0.33±0.10	0.34±0.09	0.36±0.18	0.07±0.04	0.32±0.07	0.05±0.02	0.25±0.05	0.34±0.18

Appendix

Table 22: Poisson / U-FNO. All 81 cells: mean

\pm

SD on one line (

n=200

); blue marks matched views. Epoch/cohort in row order: 130/M120, 130/M120, 137/M120, 200/M120, 200/M120, 134/M120, 200/M120, 151/M120, 163/M120.

	Test observation pattern
Train	R50	H	V	CL	BL	LI	BD	R65	R80
R50	0.01±0.00	0.05±0.03	0.05±0.03	0.47±0.19	0.81±0.24	0.04±0.02	0.88±0.24	0.15±0.04	0.29±0.08
H	0.06±0.03	0.02±0.01	0.36±0.23	2.14±1.37	2.99±1.92	0.25±0.16	3.01±1.94	0.19±0.10	0.51±0.23
V	0.03±0.01	0.18±0.09	0.02±0.01	0.65±0.31	1.21±0.56	0.12±0.06	1.26±0.58	0.11±0.05	0.25±0.13
CL	0.08±0.03	0.08±0.03	0.09±0.03	0.08±0.04	0.47±0.21	0.08±0.03	0.54±0.18	0.07±0.04	0.11±0.04
BL	0.67±0.17	0.86±0.41	0.98±0.58	0.71±0.33	0.14±0.09	0.86±0.40	0.10±0.05	0.34±0.09	0.29±0.08

Appendix

Table 23: Poisson / FNO. All 81 cells: mean

\pm

SD on one line (

n=200

); blue marks matched views. Epoch/cohort in row order: 433/INT, 500/O500, 500/O500, 262/INT, 130/M120, 480/INT, 130/M120, 500/O500, 461/INT.

	Test observation pattern
Train	R50	H	V	CL	BL	LI	BD	R65	R80
R50	0.02±0.01	0.13±0.09	0.13±0.09	1.14±0.53	1.53±0.64	0.12±0.06	1.53±0.63	0.30±0.07	1.35±0.41
H	0.02±0.01	0.01±0.00	0.06±0.02	0.43±0.21	0.87±0.31	0.04±0.01	0.92±0.31	0.15±0.03	0.34±0.08
V	0.01±0.01	0.05±0.02	0.01±0.00	0.39±0.17	0.80±0.14	0.03±0.01	0.81±0.10	0.12±0.01	0.25±0.03
CL	0.09±0.04	0.08±0.02	0.07±0.02	0.12±0.07	0.74±0.26	0.08±0.02	0.83±0.27	0.14±0.05	0.20±0.06
BL	0.64±0.35	0.68±0.31	0.65±0.29	0.81±0.26	0.12±0.07	0.62±0.31	0.09±0.04	0.65±0.40	0.66±0.41

Appendix

Table 24: Poisson / CNO. All 81 cells: mean

\pm

SD on one line (

n=200

); blue marks matched views. Epoch/cohort in row order: 275/INT, 500/O500, 500/O500, 500/O500, 500/O500, 500/O500, 500/O500, 500/O500, 500/O500.

	Test observation pattern
Train	R50	H	V	CL	BL	LI	BD	R65	R80
R50	0.02±0.01	0.16±0.04	0.15±0.05	0.59±0.11	0.89±0.09	0.13±0.03	0.91±0.09	0.06±0.02	0.19±0.04
H	0.19±0.08	0.03±0.01	0.29±0.14	0.65±0.27	0.83±0.17	0.16±0.11	0.81±0.15	0.10±0.05	0.09±0.03
V	0.17±0.06	0.24±0.09	0.03±0.01	0.54±0.19	0.78±0.16	0.11±0.07	0.76±0.16	0.12±0.06	0.10±0.06
CL	0.60±0.24	1.44±0.55	0.29±0.12	0.23±0.09	0.61±0.24	0.94±0.36	0.65±0.24	0.52±0.20	0.38±0.16
BL	7.26±2.82	13.65±7.26	11.65±5.08	1.62±0.95	0.13±0.08	11.44±5.05	0.07±0.03	6.52±2.16	4.46±1.19

Appendix

Table 25: Poisson / DeepONet. All 81 cells: mean

\pm

SD on one line (

n=200

); blue marks matched views. Epoch/cohort in row order: 500/O500, 500/O500, 500/O500, 66/INT, 500/O500, 500/O500, 500/O500, 500/O500, 500/O500.

	Test observation pattern
Train	R50	H	V	CL	BL	LI	BD	R65	R80
R50	0.01±0.00	0.02±0.01	0.02±0.01	15.43±33.93	71.17±92.13	0.02±0.01	85.57±55.23	0.01±0.00	0.01±0.00
H	0.01±0.00	0.01±0.00	0.05±0.04	6.17±17.29	26.94±23.73	0.02±0.01	33.87±19.62	0.01±0.00	0.01±0.00
V	0.02±0.01	0.08±0.04	0.01±0.01	6.43±9.06	4.12±7.29	0.06±0.03	0.92±0.38	0.01±0.00	0.01±0.00
CL	0.02±0.01	0.02±0.01	0.02±0.01	0.04±0.03	0.38±0.16	0.02±0.01	0.34±0.12	0.02±0.01	0.02±0.01
BL	0.27±0.14	0.30±0.15	0.29±0.15	0.60±0.27	0.29±0.16	0.29±0.16	0.26±0.16	0.27±0.14	0.27±0.15

Appendix

Table 26: Poisson / GNOT. All 81 cells: mean

\pm

SD on one line (

n=200

); blue marks matched views. Epoch/cohort in row order: 500/O500, 500/O500, 200/P10, 500/O500, 200/M120, 500/O500, 200/M120, 200/M120, 135/INT.

	Test observation pattern
Train	R50	H	V	CL	BL	LI	BD	R65	R80
R50	0.00±0.00	0.29±0.11	0.36±0.17	0.66±0.22	1.98±1.31	0.29±0.11	2.17±1.37	0.08±0.02	0.31±0.07
H	1.74±0.86	0.00±0.01	0.73±0.14	0.62±0.20	1.00±0.43	0.32±0.08	1.02±0.33	1.88±0.96	1.44±0.87
V	0.50±0.08	0.65±0.12	0.06±0.03	0.83±0.36	1.05±0.39	0.34±0.07	1.09±0.38	0.40±0.08	0.35±0.09
CL	0.07±0.04	0.13±0.06	0.12±0.05	0.07±0.04	0.69±0.19	0.08±0.02	0.68±0.15	0.08±0.02	0.11±0.04
BL	1.25±0.59	1.04±0.43	0.99±0.39	0.93±0.47	0.15±0.09	0.99±0.37	0.10±0.06	1.32±0.62	1.35±0.64

Appendix

Table 27: Poisson / Transolver. All 81 cells: mean

\pm

SD on one line (

n=200

); blue marks matched views. Epoch/cohort in row order: 500/O500, 500/O500, 189/INT, 500/O500, 500/O500, 191/INT, 500/O500, 500/O500, 500/O500.

	Test observation pattern
Train	R50	H	V	CL	BL	LI	BD	R65	R80
R50	0.04±0.01	0.16±0.07	0.15±0.06	0.66±0.16	0.95±0.20	0.14±0.05	1.05±0.18	0.28±0.05	0.55±0.14
H	0.14±0.08	0.01±0.01	0.23±0.08	1.05±0.57	1.73±1.19	0.16±0.07	2.25±1.45	0.30±0.07	0.52±0.10
V	0.14±0.08	0.25±0.12	0.02±0.01	1.31±0.78	1.68±0.83	0.16±0.08	2.02±1.00	0.29±0.08	0.52±0.12
CL	0.38±0.03	0.37±0.05	0.38±0.06	0.13±0.08	0.45±0.16	0.37±0.04	0.45±0.18	0.24±0.03	0.11±0.03
BL	0.76±0.08	0.63±0.20	0.83±0.23	0.83±0.21	0.07±0.05	0.61±0.19	0.05±0.03	0.69±0.09	0.64±0.10

Appendix

Table 28: Poisson / PINO. All 81 cells: mean

\pm

SD on one line (

n=200

); blue marks matched views. Epoch/cohort in row order: 500/O500, 500/O500, 500/O500, 130/M120, 500/O500, 500/O500, 500/O500, 500/O500, 500/O500.

	Test observation pattern
Train	R50	H	V	CL	BL	LI	BD	R65	R80
R50	1.07±0.29	1.01±0.27	1.02±0.28	1.14±0.37	1.37±0.51	1.03±0.28	1.34±0.51	0.95±0.26	0.45±0.11
H	0.47±0.01	0.02±0.01	0.68±0.06	0.52±0.16	0.76±0.23	0.26±0.03	0.80±0.20	0.44±0.01	0.40±0.02
V	0.47±0.02	0.67±0.06	0.03±0.01	0.60±0.18	0.78±0.27	0.28±0.05	0.81±0.25	0.46±0.02	0.47±0.03
CL	0.04±0.02	0.09±0.03	0.12±0.07	0.07±0.03	0.38±0.17	0.08±0.03	0.33±0.13	0.06±0.02	0.08±0.03
BL	0.63±0.43	1.36±0.47	1.42±0.48	0.29±0.10	0.06±0.03	1.24±0.52	0.05±0.02	0.52±0.37	0.27±0.21

Appendix

Table 29: Helmholtz / U-FNO. All 81 cells: mean

\pm

SD on one line (

n=200

); blue marks matched views. Epoch/cohort in row order: 132/M120, 133/M120, 135/M120, 138/M120, 200/M120, 134/M120, 200/M120, 155/M120, 500/O500.

	Test observation pattern
Train	R50	H	V	CL	BL	LI	BD	R65	R80
R50	0.01±0.00	0.07±0.04	0.08±0.05	1.16±0.41	1.60±0.65	0.05±0.04	1.59±0.65	0.13±0.03	0.44±0.10
H	0.02±0.00	0.01±0.00	0.12±0.05	0.70±0.29	1.25±0.40	0.08±0.03	1.29±0.39	0.27±0.09	0.59±0.20
V	0.03±0.01	0.19±0.08	0.01±0.00	0.68±0.22	1.11±0.29	0.12±0.05	1.12±0.28	0.10±0.02	0.26±0.07
CL	0.04±0.00	0.04±0.01	0.04±0.01	0.04±0.02	0.33±0.17	0.04±0.01	0.42±0.21	0.07±0.02	0.13±0.04
BL	0.53±0.14	0.57±0.20	0.70±0.32	0.79±0.28	0.06±0.03	0.55±0.18	0.04±0.02	0.56±0.15	0.69±0.19

Appendix

Table 30: Helmholtz / FNO. All 81 cells: mean

\pm

SD on one line (

n=200

); blue marks matched views. All training views: 500/O500.

	Test observation pattern
Train	R50	H	V	CL	BL	LI	BD	R65	R80
R50	0.00±0.00	0.07±0.05	0.08±0.05	0.49±0.15	0.85±0.09	0.07±0.03	0.90±0.06	0.11±0.01	0.22±0.04
H	0.02±0.01	0.01±0.00	0.09±0.03	0.42±0.21	0.82±0.23	0.05±0.01	0.86±0.22	0.20±0.03	0.43±0.07
V	0.02±0.00	0.07±0.02	0.01±0.00	0.44±0.20	0.85±0.24	0.04±0.01	0.90±0.23	0.12±0.02	0.20±0.04
CL	0.15±0.05	0.15±0.05	0.14±0.04	0.09±0.05	0.57±0.22	0.14±0.05	0.63±0.16	0.17±0.04	0.22±0.05
BL	0.95±0.34	1.00±0.34	1.08±0.40	0.74±0.30	0.09±0.05	1.00±0.34	0.07±0.03	0.90±0.32	0.90±0.32

Appendix

Table 31: Helmholtz / CNO. All 81 cells: mean

\pm

SD on one line (

n=200

); blue marks matched views. All training views: 500/O500.

	Test observation pattern
Train	R50	H	V	CL	BL	LI	BD	R65	R80
R50	0.02±0.01	0.14±0.04	0.13±0.05	0.63±0.13	0.92±0.10	0.11±0.03	0.93±0.09	0.08±0.03	0.17±0.06
H	0.22±0.03	0.02±0.01	0.34±0.08	0.53±0.16	0.84±0.14	0.18±0.08	0.81±0.12	0.11±0.02	0.04±0.01
V	0.31±0.09	0.38±0.14	0.02±0.01	0.51±0.16	0.82±0.22	0.19±0.09	0.82±0.23	0.30±0.09	0.30±0.09
CL	0.28±0.09	3.01±0.82	3.12±1.11	0.16±0.06	0.78±0.28	1.43±0.50	0.89±0.19	0.23±0.06	0.23±0.08
BL	7.60±3.38	15.28±7.37	13.81±6.76	1.60±0.80	0.10±0.07	12.74±5.56	0.06±0.02	6.77±3.09	4.59±2.21

Appendix

Table 32: Helmholtz / DeepONet. All 81 cells: mean

\pm

SD on one line (

n=200

); blue marks matched views. All training views: 500/O500.

	Test observation pattern
Train	R50	H	V	CL	BL	LI	BD	R65	R80
R50	0.01±0.00	0.04±0.12	0.05±0.33	132.35±117.02	81.74±74.92	0.02±0.01	13.44±11.45	0.01±0.00	0.00±0.00
H	0.01±0.00	0.01±0.00	0.07±0.14	65.50±98.78	62.51±54.10	0.03±0.01	51.71±16.29	0.01±0.00	0.01±0.00
V	0.01±0.00	0.16±0.44	0.01±0.00	20.08±37.63	13.43±8.04	0.03±0.01	11.85±4.28	0.01±0.00	0.01±0.00
CL	0.02±0.01	0.02±0.01	0.02±0.01	0.03±0.02	0.49±0.13	0.02±0.01	0.49±0.13	0.02±0.01	0.02±0.01
BL	0.28±0.11	0.29±0.11	0.29±0.11	0.51±0.39	0.20±0.15	0.29±0.11	0.16±0.12	0.29±0.11	0.28±0.11

Appendix

Table 33: Helmholtz / GNOT. All 81 cells: mean

\pm

SD on one line (

n=200

); blue marks matched views. Epoch/cohort in row order: 500/O500, 500/O500, 500/O500, 500/O500, 200/M120, 45/P10, 500/O500, 200/P10, 500/O500.

	Test observation pattern
Train	R50	H	V	CL	BL	LI	BD	R65	R80
R50	0.00±0.00	0.31±0.10	0.31±0.13	0.77±0.21	1.37±0.39	0.30±0.09	1.38±0.37	0.07±0.01	0.12±0.03
H	0.58±0.05	0.00±0.00	0.81±0.22	0.73±0.33	1.93±0.89	0.42±0.05	1.91±0.77	0.54±0.11	0.48±0.25
V	0.49±0.07	0.71±0.08	0.00±0.00	0.55±0.16	0.83±0.21	0.34±0.06	0.84±0.19	0.46±0.08	0.46±0.10
CL	0.34±0.12	0.56±0.19	0.46±0.17	0.05±0.04	0.60±0.20	0.55±0.20	0.60±0.21	0.30±0.11	0.37±0.11
BL	0.56±0.15	0.52±0.15	0.53±0.16	0.58±0.20	0.07±0.05	0.49±0.14	0.05±0.03	0.64±0.17	0.71±0.20

Appendix

Table 34: Helmholtz / Transolver. All 81 cells: mean

\pm

SD on one line (

n=200

); blue marks matched views. All training views: 500/O500.

	Test observation pattern
Train	R50	H	V	CL	BL	LI	BD	R65	R80
R50	0.04±0.01	0.12±0.04	0.11±0.05	0.56±0.18	0.98±0.17	0.10±0.04	1.12±0.15	0.20±0.03	0.36±0.05
H	0.09±0.04	0.02±0.01	0.27±0.13	0.79±0.15	1.03±0.20	0.18±0.08	1.14±0.23	0.18±0.03	0.32±0.04
V	0.11±0.04	0.27±0.11	0.02±0.01	0.77±0.16	1.22±0.30	0.19±0.08	1.28±0.32	0.19±0.03	0.34±0.04
CL	0.28±0.03	0.28±0.05	0.27±0.06	0.11±0.05	0.44±0.16	0.27±0.04	0.44±0.17	0.18±0.02	0.11±0.02
BL	0.69±0.11	0.67±0.12	0.74±0.18	0.81±0.15	0.08±0.06	0.65±0.16	0.06±0.03	0.64±0.12	0.61±0.13

Appendix

Table 35: Helmholtz / PINO. All 81 cells: mean

\pm

SD on one line (

n=200

); blue marks matched views. All training views: 500/O500.

	Test observation pattern
Train	R50	H	V	CL	BL	LI	BD	R65	R80
R50	0.36±0.07	0.31±0.06	0.33±0.06	0.55±0.10	0.61±0.10	0.32±0.06	0.60±0.11	0.19±0.05	0.10±0.02
H	0.33±0.02	0.35±0.03	0.42±0.03	0.61±0.09	0.67±0.10	0.37±0.03	0.65±0.10	0.15±0.02	0.06±0.04
V	0.15±0.04	0.26±0.05	0.07±0.03	0.89±0.25	0.93±0.19	0.16±0.04	0.92±0.19	0.14±0.05	0.10±0.06
CL	0.13±0.07	0.13±0.06	0.14±0.07	0.15±0.06	0.14±0.05	0.13±0.06	0.14±0.05	0.08±0.06	0.09±0.04
BL	0.31±0.04	0.27±0.05	0.28±0.05	0.24±0.07	0.11±0.03	0.27±0.05	0.11±0.03	0.17±0.05	0.11±0.05

Appendix

Table 36: Heat / U-FNO. All 81 cells: mean

\pm

SD on one line (

n=200

); blue marks matched views. Epoch/cohort in row order: 500/R500, 500/R500, 500/R500, 500/R500, 500/O500, 500/R500, 500/R500, 500/R500, 500/R500.

	Test observation pattern
Train	R50	H	V	CL	BL	LI	BD	R65	R80
R50	0.05±0.05	0.06±0.04	0.06±0.04	0.47±0.17	0.56±0.16	0.05±0.05	0.56±0.17	0.11±0.04	0.08±0.04
H	0.05±0.05	0.05±0.05	0.07±0.04	0.42±0.13	0.49±0.10	0.06±0.04	0.49±0.11	0.05±0.05	0.06±0.04
V	0.05±0.04	0.07±0.04	0.05±0.04	0.38±0.11	0.46±0.11	0.06±0.04	0.45±0.11	0.05±0.04	0.06±0.04
CL	0.07±0.06	0.07±0.05	0.07±0.04	0.07±0.04	0.08±0.03	0.06±0.05	0.08±0.03	0.11±0.05	0.08±0.06
BL	0.08±0.03	0.11±0.04	0.11±0.04	0.17±0.05	0.06±0.05	0.10±0.03	0.06±0.05	0.07±0.03	0.06±0.04

Appendix

Table 37: Heat / FNO. All 81 cells: mean

\pm

SD on one line (

n=200

); blue marks matched views. Epoch/cohort in row order: 500/O500, 500/O500, 500/O500, 500/O500, 500/O500, 500/INT, 500/O500, 500/O500, 500/O500.

	Test observation pattern
Train	R50	H	V	CL	BL	LI	BD	R65	R80
R50	0.06±0.04	0.18±0.09	0.21±0.10	0.62±0.09	0.66±0.08	0.19±0.07	0.65±0.08	0.06±0.04	0.06±0.04
H	0.05±0.05	0.05±0.05	0.14±0.07	0.46±0.12	0.59±0.10	0.08±0.03	0.59±0.11	0.05±0.04	0.05±0.04
V	0.06±0.04	0.18±0.08	0.06±0.05	0.48±0.11	0.55±0.10	0.13±0.06	0.54±0.10	0.06±0.04	0.06±0.05
CL	0.09±0.04	0.09±0.05	0.08±0.04	0.12±0.05	0.30±0.12	0.08±0.04	0.33±0.13	0.08±0.04	0.07±0.05
BL	0.10±0.04	0.14±0.04	0.18±0.06	0.54±0.16	0.10±0.04	0.15±0.04	0.09±0.03	0.08±0.04	0.07±0.04

Appendix

Table 38: Heat / CNO. All 81 cells: mean

\pm

SD on one line (

n=200

); blue marks matched views. All training views: 500/O500.

	Test observation pattern
Train	R50	H	V	CL	BL	LI	BD	R65	R80
R50	0.08±0.03	0.52±0.13	0.52±0.13	1.04±0.19	1.12±0.21	0.46±0.10	1.02±0.17	0.21±0.09	0.12±0.07
H	0.26±0.07	0.09±0.03	0.85±0.40	0.63±0.16	0.71±0.12	0.50±0.30	0.64±0.12	0.29±0.13	0.28±0.12
V	0.49±0.13	1.06±0.42	0.09±0.04	0.65±0.14	0.65±0.12	0.61±0.25	0.61±0.11	0.67±0.20	0.53±0.18
CL	0.25±0.11	1.06±1.38	8.75±10.37	0.20±0.07	0.49±0.15	1.18±1.76	0.46±0.14	0.24±0.06	0.21±0.06
BL	6.02±4.26	17.45±9.95	17.62±10.32	0.70±0.29	0.11±0.05	16.67±10.00	0.10±0.04	4.37±3.08	1.84±1.00

Appendix

Table 39: Heat / DeepONet. All 81 cells: mean

\pm

SD on one line (

n=200

); blue marks matched views. Epoch/cohort in row order: 500/O500, 500/O500, 500/O500, 500/R500, 500/R500, 500/O500, 500/R500, 500/R500, 500/O500.

	Test observation pattern
Train	R50	H	V	CL	BL	LI	BD	R65	R80
R50	0.07±0.03	0.10±0.03	0.10±0.03	0.51±0.17	0.48±0.16	0.10±0.03	0.50±0.16	0.07±0.03	0.07±0.03
H	0.09±0.04	0.10±0.03	0.15±0.05	3.38±3.97	4.59±4.14	0.12±0.04	3.56±3.87	0.09±0.04	0.09±0.04
V	0.09±0.03	0.12±0.04	0.09±0.03	0.74±0.71	0.63±0.24	0.10±0.03	0.63±0.20	0.09±0.03	0.09±0.03
CL	0.07±0.05	0.08±0.05	0.08±0.05	0.13±0.05	0.23±0.08	0.08±0.04	0.29±0.08	0.07±0.05	0.07±0.05
BL	0.08±0.05	0.10±0.04	0.10±0.04	0.34±0.11	0.10±0.04	0.10±0.04	0.10±0.04	0.08±0.05	0.08±0.05

Appendix

Table 40: Heat / GNOT. All 81 cells: mean

\pm

SD on one line (

n=200

); blue marks matched views. Epoch/cohort in row order: 200/P10, 200/F200, 200/F200, 200/F200, 200/F200, 200/F200, 200/F200, 200/P10, 200/F200.

	Test observation pattern
Train	R50	H	V	CL	BL	LI	BD	R65	R80
R50	0.05±0.04	0.19±0.08	0.22±0.09	0.74±0.14	0.82±0.10	0.17±0.05	0.82±0.10	0.06±0.04	0.07±0.03
H	0.26±0.04	0.07±0.05	0.48±0.06	0.50±0.14	0.53±0.14	0.16±0.03	0.52±0.13	0.19±0.04	0.14±0.04
V	0.22±0.02	0.43±0.04	0.07±0.04	0.74±0.15	0.82±0.14	0.21±0.04	0.82±0.14	0.15±0.03	0.10±0.03
CL	0.14±0.02	0.17±0.03	0.17±0.03	0.35±0.11	0.51±0.10	0.16±0.02	0.50±0.11	0.13±0.02	0.11±0.02
BL	0.49±0.04	0.35±0.08	0.34±0.07	0.45±0.11	0.08±0.04	0.35±0.07	0.08±0.04	0.49±0.03	0.40±0.02

Appendix

Table 41: Heat / Transolver. All 81 cells: mean

\pm

SD on one line (

n=200

); blue marks matched views. Epoch/cohort in row order: 257/INT, 500/O500, 124/INT, 130/M120, 200/M120, 500/O500, 500/O500, 157/M120, 133/M120.

	Test observation pattern
Train	R50	H	V	CL	BL	LI	BD	R65	R80
R50	0.06±0.05	0.07±0.06	0.07±0.06	0.55±0.11	0.66±0.10	0.07±0.04	0.66±0.10	0.06±0.04	0.10±0.03
H	0.06±0.03	0.05±0.04	0.12±0.03	0.44±0.11	0.55±0.09	0.09±0.02	0.55±0.09	0.06±0.03	0.07±0.03
V	0.07±0.04	0.15±0.04	0.06±0.04	0.54±0.11	0.63±0.08	0.11±0.03	0.63±0.08	0.06±0.05	0.07±0.05
CL	0.08±0.03	0.09±0.02	0.17±0.07	0.08±0.03	0.09±0.03	0.12±0.05	0.09±0.03	0.08±0.03	0.09±0.04
BL	0.08±0.04	0.18±0.05	0.16±0.04	0.29±0.09	0.07±0.04	0.12±0.04	0.07±0.05	0.09±0.05	0.10±0.05

Appendix

Table 42: Heat / PINO. All 81 cells: mean

\pm

SD on one line (

n=200

); blue marks matched views. Epoch/cohort in row order: 500/O500, 500/R500, 500/O500, 500/O500, 500/O500, 500/R500, 500/R500, 500/O500, 500/O500.

	Test observation pattern
Train	R50	H	V	CL	BL	LI	BD	R65	R80
R50	0.16±0.02	0.18±0.03	0.18±0.03	0.47±0.10	0.50±0.10	0.17±0.02	0.49±0.10	0.10±0.02	0.08±0.03
H	0.08±0.03	0.09±0.02	0.12±0.03	0.41±0.12	0.43±0.11	0.08±0.03	0.42±0.11	0.07±0.03	0.07±0.03
V	0.12±0.02	0.19±0.02	0.11±0.01	0.39±0.09	0.40±0.08	0.12±0.02	0.39±0.08	0.09±0.02	0.08±0.03
CL	0.08±0.03	0.09±0.03	0.09±0.04	0.18±0.05	0.26±0.05	0.08±0.03	0.24±0.05	0.08±0.05	0.08±0.06
BL	0.26±0.03	0.22±0.05	0.23±0.05	0.34±0.07	0.25±0.04	0.21±0.05	0.24±0.04	0.12±0.04	0.09±0.04

Appendix

Table 43: Reaction–diffusion / U-FNO. All 81 cells: mean

\pm

SD on one line (

n=200

); blue marks matched views. Epoch/cohort in row order: 500/O500, 500/R500, 500/R500, 500/R500, 500/R500, 500/R500, 500/R500, 500/O500, 500/O500.

	Test observation pattern
Train	R50	H	V	CL	BL	LI	BD	R65	R80
R50	0.05±0.04	0.06±0.04	0.06±0.04	0.38±0.10	0.44±0.09	0.06±0.04	0.44±0.09	0.07±0.03	0.09±0.03
H	0.06±0.04	0.05±0.04	0.06±0.03	0.32±0.09	0.43±0.11	0.06±0.04	0.43±0.11	0.06±0.04	0.08±0.03
V	0.06±0.04	0.07±0.03	0.05±0.04	0.34±0.09	0.40±0.10	0.06±0.04	0.40±0.10	0.06±0.03	0.07±0.03
CL	0.09±0.06	0.08±0.05	0.08±0.06	0.08±0.04	0.09±0.04	0.08±0.06	0.09±0.03	0.10±0.06	0.10±0.06
BL	0.12±0.04	0.12±0.04	0.15±0.05	0.21±0.07	0.07±0.04	0.12±0.04	0.08±0.04	0.10±0.04	0.09±0.03

Appendix

Table 44: Reaction–diffusion / FNO. All 81 cells: mean

\pm

SD on one line (

n=200

); blue marks matched views. Epoch/cohort in row order: 500/O500, 500/O500, 500/O500, 469/INT, 500/O500, 500/O500, 500/O500, 500/O500, 500/O500.

	Test observation pattern
Train	R50	H	V	CL	BL	LI	BD	R65	R80
R50	0.05±0.04	0.09±0.04	0.10±0.05	0.56±0.09	0.60±0.09	0.09±0.03	0.60±0.09	0.05±0.04	0.06±0.04
H	0.06±0.04	0.06±0.04	0.11±0.05	0.49±0.11	0.56±0.10	0.07±0.03	0.56±0.10	0.07±0.05	0.07±0.06
V	0.06±0.04	0.13±0.06	0.06±0.04	0.56±0.13	0.62±0.12	0.08±0.03	0.63±0.13	0.06±0.04	0.06±0.04
CL	0.08±0.03	0.08±0.03	0.08±0.03	0.11±0.04	0.28±0.12	0.08±0.03	0.32±0.12	0.07±0.03	0.07±0.03
BL	0.09±0.02	0.16±0.04	0.15±0.03	0.55±0.16	0.10±0.04	0.14±0.02	0.09±0.03	0.08±0.03	0.07±0.03

Appendix

Table 45: Reaction–diffusion / CNO. All 81 cells: mean

\pm

SD on one line (

n=200

); blue marks matched views. All training views: 500/O500.

	Test observation pattern
Train	R50	H	V	CL	BL	LI	BD	R65	R80
R50	0.10±0.03	0.62±0.17	0.70±0.14	2.42±0.63	2.53±0.55	0.44±0.10	2.24±0.19	0.11±0.03	0.10±0.03
H	0.33±0.03	0.10±0.03	0.55±0.08	0.65±0.12	0.76±0.11	0.23±0.05	0.65±0.09	0.21±0.03	0.14±0.03
V	0.29±0.03	0.48±0.09	0.11±0.03	0.58±0.11	0.66±0.11	0.22±0.04	0.59±0.08	0.19±0.03	0.12±0.03
CL	0.27±0.06	0.75±0.82	0.45±0.25	0.20±0.07	0.43±0.11	0.34±0.17	0.42±0.10	0.27±0.06	0.23±0.05
BL	4.56±2.63	10.29±4.56	16.34±7.40	0.69±0.42	0.12±0.04	13.20±5.71	0.10±0.04	3.78±2.20	1.72±1.22

Appendix

Table 46: Reaction–diffusion / DeepONet. All 81 cells: mean

\pm

SD on one line (

n=200

); blue marks matched views. Epoch/cohort in row order: 500/O500, 500/O500, 500/O500, 500/O500, 500/O500, 500/O500, 500/O500, 251/INT, 500/O500.

	Test observation pattern
Train	R50	H	V	CL	BL	LI	BD	R65	R80
R50	0.09±0.03	0.12±0.04	0.32±1.50	11.15±7.82	18.08±2.25	0.12±0.04	18.48±0.52	0.09±0.03	0.09±0.03
H	0.10±0.04	0.11±0.04	0.14±0.05	8.92±7.86	26.61±15.88	0.12±0.04	33.23±15.23	0.10±0.04	0.10±0.04
V	0.10±0.03	0.15±0.06	0.10±0.03	0.66±0.20	0.72±0.19	0.12±0.04	0.68±0.18	0.09±0.03	0.09±0.03
CL	0.14±0.04	0.14±0.04	0.14±0.04	0.22±0.07	0.34±0.11	0.14±0.04	0.38±0.10	0.14±0.04	0.14±0.04
BL	0.10±0.05	0.11±0.05	0.11±0.05	0.33±0.10	0.11±0.04	0.11±0.05	0.10±0.04	0.10±0.05	0.10±0.05

Appendix

Table 47: Reaction–diffusion / GNOT. All 81 cells: mean

\pm

SD on one line (

n=200

); blue marks matched views. Epoch/cohort in row order: 200/P10, 200/F200, 200/F200, 200/F200, 200/F200, 200/F200, 200/F200, 200/F200, 200/F200.

	Test observation pattern
Train	R50	H	V	CL	BL	LI	BD	R65	R80
R50	0.07±0.05	0.15±0.05	0.14±0.05	0.54±0.10	0.65±0.08	0.13±0.03	0.65±0.08	0.08±0.05	0.08±0.04
H	0.20±0.02	0.07±0.05	0.56±0.04	0.58±0.10	0.64±0.11	0.21±0.03	0.64±0.11	0.14±0.03	0.11±0.03
V	0.17±0.02	0.30±0.04	0.11±0.03	0.61±0.11	0.68±0.11	0.19±0.03	0.68±0.11	0.14±0.02	0.12±0.02
CL	0.12±0.03	0.10±0.03	0.11±0.03	0.18±0.09	0.40±0.14	0.10±0.03	0.46±0.16	0.12±0.03	0.10±0.03
BL	0.23±0.03	0.22±0.04	0.21±0.04	0.41±0.13	0.17±0.07	0.22±0.03	0.15±0.04	0.19±0.02	0.19±0.02

Appendix

Table 48: Reaction–diffusion / Transolver. All 81 cells: mean

\pm

SD on one line (

n=200

); blue marks matched views. Epoch/cohort in row order: 200/P10, 200/M120, 200/F200, 200/M120, 35/INT, 47/INT, 200/F200, 200/F200, 200/F200.

	Test observation pattern
Train	R50	H	V	CL	BL	LI	BD	R65	R80
R50	0.06±0.04	0.09±0.04	0.08±0.04	0.49±0.10	0.57±0.10	0.08±0.04	0.57±0.10	0.06±0.04	0.63±0.04
H	0.10±0.06	0.06±0.04	0.12±0.04	0.47±0.09	0.58±0.08	0.10±0.04	0.58±0.09	0.10±0.06	0.11±0.05
V	0.08±0.02	0.13±0.04	0.06±0.04	0.46±0.09	0.55±0.08	0.10±0.04	0.56±0.08	0.08±0.02	0.09±0.04
CL	0.08±0.02	0.08±0.03	0.08±0.03	0.08±0.03	0.08±0.03	0.07±0.03	0.08±0.03	0.08±0.02	0.08±0.02
BL	0.08±0.03	0.16±0.05	0.16±0.05	0.27±0.08	0.07±0.04	0.11±0.03	0.07±0.04	0.14±0.03	0.26±0.03

Appendix

Table 49: Reaction–diffusion / PINO. All 81 cells: mean

\pm

SD on one line (

n=200

); blue marks matched views. Epoch/cohort in row order: 500/O500, 500/O500, 500/R500, 500/R500, 500/R500, 500/R500, 500/O500, 392/INT, 391/INT.

	Test observation pattern
Train	R50	H	V	CL	BL	LI	BD	R65	R80
R50	0.25±0.01	0.30±0.03	0.30±0.03	0.76±0.16	0.80±0.17	0.28±0.02	0.81±0.16	0.11±0.02	0.08±0.02
H	0.91±0.15	0.89±0.12	1.05±0.19	1.20±0.25	1.23±0.33	0.97±0.14	1.21±0.34	0.11±0.04	0.16±0.05
V	0.16±0.03	0.24±0.04	0.19±0.04	0.44±0.10	0.41±0.10	0.20±0.04	0.42±0.09	0.09±0.01	0.10±0.01
CL	0.07±0.03	0.07±0.03	0.07±0.03	0.09±0.03	0.13±0.04	0.07±0.03	0.12±0.05	0.07±0.03	0.07±0.03
BL	0.16±0.05	0.17±0.06	0.17±0.06	0.23±0.06	0.13±0.04	0.16±0.05	0.12±0.04	0.14±0.05	0.14±0.05

Appendix

Table 50: Burgers / U-FNO. All 81 cells: mean

\pm

SD on one line (

n=200

); blue marks matched views. All training views: 500/R500.

	Test observation pattern
Train	R50	H	V	CL	BL	LI	BD	R65	R80
R50	0.04±0.03	0.05±0.03	0.06±0.03	0.42±0.12	0.48±0.11	0.05±0.03	0.49±0.11	0.06±0.03	0.08±0.02
H	0.05±0.03	0.04±0.03	0.08±0.03	0.48±0.12	0.54±0.11	0.06±0.03	0.55±0.11	0.06±0.03	0.07±0.02
V	0.05±0.03	0.06±0.03	0.05±0.03	0.32±0.10	0.37±0.09	0.06±0.03	0.38±0.09	0.06±0.03	0.06±0.03
CL	0.06±0.04	0.06±0.04	0.05±0.04	0.07±0.03	0.08±0.03	0.06±0.04	0.08±0.04	0.06±0.04	0.06±0.04
BL	0.07±0.03	0.10±0.04	0.12±0.05	0.25±0.09	0.05±0.03	0.11±0.04	0.05±0.03	0.07±0.03	0.07±0.02

Appendix

Table 51: Burgers / FNO. All 81 cells: mean

\pm

SD on one line (

n=200

); blue marks matched views. Epoch/cohort in row order: 500/R500, 500/O500, 500/R500, 500/R500, 500/R500, 500/R500, 500/R500, 500/R500, 500/R500.

	Test observation pattern
Train	R50	H	V	CL	BL	LI	BD	R65	R80
R50	0.05±0.03	0.11±0.07	0.12±0.07	0.60±0.13	0.70±0.14	0.11±0.03	0.71±0.13	0.05±0.03	0.05±0.03
H	0.08±0.03	0.05±0.03	0.21±0.09	0.56±0.13	0.65±0.11	0.12±0.03	0.66±0.12	0.07±0.03	0.06±0.03
V	0.08±0.03	0.20±0.08	0.05±0.03	0.51±0.12	0.58±0.10	0.13±0.04	0.58±0.09	0.07±0.03	0.06±0.03
CL	0.07±0.02	0.08±0.03	0.08±0.03	0.13±0.06	0.32±0.13	0.08±0.03	0.36±0.13	0.07±0.02	0.07±0.03
BL	0.13±0.02	0.18±0.03	0.17±0.04	0.61±0.17	0.10±0.04	0.15±0.03	0.08±0.03	0.09±0.02	0.07±0.02

Appendix

Table 52: Burgers / CNO. All 81 cells: mean

\pm

SD on one line (

n=200

); blue marks matched views. All training views: 500/O500.

	Test observation pattern
Train	R50	H	V	CL	BL	LI	BD	R65	R80
R50	0.10±0.04	0.64±0.11	0.51±0.13	0.97±0.10	1.02±0.08	0.56±0.08	1.02±0.06	0.10±0.04	0.11±0.04
H	0.94±0.21	0.11±0.04	2.60±1.43	0.70±0.14	0.69±0.12	1.34±0.60	0.62±0.10	0.81±0.18	0.61±0.14
V	0.89±0.21	1.67±0.45	0.11±0.04	0.65±0.12	0.66±0.18	0.98±0.31	0.59±0.18	0.73±0.19	0.52±0.14
CL	0.32±0.07	1.35±0.90	0.70±0.30	0.20±0.06	0.43±0.12	0.48±0.19	0.40±0.11	0.33±0.07	0.28±0.06
BL	2.86±1.33	7.00±3.12	5.34±2.62	0.68±0.28	0.14±0.05	5.74±2.60	0.12±0.04	2.21±1.08	1.11±0.62

Appendix

Table 53: Burgers / DeepONet. All 81 cells: mean

\pm

SD on one line (

n=200

); blue marks matched views. Epoch/cohort in row order: 500/O500, 500/R500, 339/INT, 500/R500, 500/O500, 443/INT, 500/O500, 491/INT, 500/O500.

	Test observation pattern
Train	R50	H	V	CL	BL	LI	BD	R65	R80
R50	0.07±0.03	0.10±0.05	0.10±0.03	0.88±0.60	0.79±0.40	0.09±0.03	0.74±0.50	0.07±0.03	0.07±0.03
H	0.08±0.03	0.07±0.03	0.11±0.04	2.15±2.62	1.07±1.11	0.09±0.03	0.72±0.29	0.08±0.03	0.08±0.03
V	0.07±0.03	0.09±0.03	0.07±0.03	0.37±0.13	0.39±0.13	0.08±0.03	0.40±0.12	0.07±0.03	0.07±0.03
CL	0.09±0.04	0.10±0.04	0.10±0.04	0.14±0.05	0.25±0.08	0.10±0.04	0.28±0.08	0.09±0.04	0.09±0.04
BL	0.10±0.04	0.13±0.05	0.12±0.04	0.41±0.14	0.12±0.05	0.12±0.04	0.12±0.05	0.10±0.04	0.10±0.04

Appendix

Table 54: Burgers / GNOT. All 81 cells: mean

\pm

SD on one line (

n=200

); blue marks matched views. Epoch/cohort in row order: 200/F200, 500/O500, 200/F200, 200/F200, 500/O500, 200/P10, 200/F200, 200/F200, 200/F200.

	Test observation pattern
Train	R50	H	V	CL	BL	LI	BD	R65	R80
R50	0.20±0.03	0.42±0.06	0.40±0.04	0.71±0.12	0.77±0.12	0.40±0.04	0.77±0.11	0.19±0.03	0.19±0.03
H	0.15±0.02	0.06±0.04	0.31±0.04	0.49±0.13	0.55±0.12	0.16±0.02	0.55±0.12	0.12±0.02	0.10±0.02
V	0.14±0.02	0.29±0.03	0.06±0.03	0.50±0.14	0.60±0.15	0.14±0.03	0.60±0.15	0.11±0.02	0.09±0.02
CL	0.07±0.03	0.08±0.03	0.09±0.03	0.10±0.05	0.26±0.11	0.07±0.03	0.28±0.10	0.06±0.03	0.07±0.03
BL	0.26±0.04	0.23±0.04	0.24±0.04	0.44±0.15	0.08±0.04	0.24±0.05	0.07±0.04	0.17±0.03	0.12±0.02

Appendix

Table 55: Burgers / Transolver. All 81 cells: mean

\pm

SD on one line (

n=200

); blue marks matched views. Epoch/cohort in row order: 200/F200, 200/M120, 200/M120, 200/M120, 200/M120, 200/M120, 73/P10, 200/F200, 200/F200.

	Test observation pattern
Train	R50	H	V	CL	BL	LI	BD	R65	R80
R50	0.05±0.04	0.06±0.03	0.06±0.03	0.44±0.12	0.50±0.12	0.06±0.03	0.51±0.11	0.05±0.03	0.06±0.03
H	0.07±0.03	0.06±0.04	0.15±0.04	0.49±0.10	0.57±0.10	0.11±0.04	0.58±0.10	0.06±0.03	0.07±0.03
V	0.13±0.07	0.14±0.04	0.05±0.04	0.50±0.10	0.55±0.10	0.11±0.03	0.55±0.09	0.16±0.09	0.17±0.09
CL	0.07±0.03	0.08±0.03	0.08±0.03	0.08±0.03	0.09±0.03	0.07±0.03	0.09±0.03	0.06±0.03	0.06±0.03
BL	0.09±0.04	0.17±0.06	0.16±0.05	0.29±0.09	0.06±0.03	0.12±0.04	0.06±0.03	0.10±0.04	0.10±0.04

Appendix

Table 56: Burgers / PINO. All 81 cells: mean

\pm

SD on one line (

n=200

); blue marks matched views. Epoch/cohort in row order: 500/R500, 500/O500, 500/R500, 500/R500, 500/O500, 500/R500, 500/R500, 500/O500, 500/O500.

	Test observation pattern
Train	R50	H	V	CL	BL	LI	BD	R65	R80
R50	0.01±0.00	0.04±0.02	0.04±0.02	0.45±0.10	0.50±0.10	0.04±0.01	0.50±0.10	0.01±0.00	0.01±0.00
H	0.04±0.00	0.01±0.00	0.07±0.01	0.32±0.10	0.34±0.09	0.03±0.00	0.34±0.09	0.03±0.01	0.03±0.01
V	0.04±0.00	0.11±0.02	0.01±0.00	0.34±0.10	0.37±0.09	0.06±0.01	0.37±0.09	0.03±0.00	0.02±0.00
CL	0.05±0.01	0.05±0.01	0.05±0.01	0.09±0.04	0.14±0.05	0.05±0.01	0.14±0.05	0.05±0.01	0.03±0.01
BL	0.22±0.03	0.24±0.05	0.24±0.05	0.23±0.07	0.05±0.02	0.23±0.04	0.05±0.02	0.07±0.01	0.05±0.01

Appendix

Table 57: Navier–Stokes / U-FNO. All 81 cells: mean

\pm

SD on one line (

n=200

); blue marks matched views. Epoch/cohort in row order: 500/R500, 500/R500, 500/R500, 500/R500, 500/R500, 500/R500, 500/O500, 500/R500, 500/R500.

	Test observation pattern
Train	R50	H	V	CL	BL	LI	BD	R65	R80
R50	0.00±0.00	0.01±0.01	0.01±0.01	0.36±0.10	0.44±0.10	0.01±0.01	0.45±0.10	0.03±0.01	0.07±0.02
H	0.01±0.00	0.00±0.00	0.05±0.02	0.42±0.12	0.55±0.13	0.04±0.02	0.55±0.13	0.02±0.00	0.03±0.01
V	0.03±0.01	0.09±0.03	0.03±0.01	0.44±0.10	0.51±0.08	0.06±0.02	0.51±0.08	0.05±0.01	0.05±0.01
CL	0.01±0.00	0.02±0.00	0.02±0.00	0.02±0.01	0.05±0.02	0.01±0.00	0.05±0.02	0.02±0.01	0.04±0.02
BL	0.06±0.01	0.09±0.02	0.14±0.08	0.30±0.09	0.01±0.00	0.10±0.03	0.01±0.00	0.05±0.02	0.04±0.02

Appendix

Table 58: Navier–Stokes / FNO. All 81 cells: mean

\pm

SD on one line (

n=200

); blue marks matched views. Epoch/cohort in row order: 500/O500, 500/O500, 36/INT, 500/O500, 500/O500, 500/O500, 500/O500, 500/O500, 500/O500.

	Test observation pattern
Train	R50	H	V	CL	BL	LI	BD	R65	R80
R50	0.01±0.01	0.11±0.10	0.11±0.09	0.96±0.22	1.00±0.24	0.10±0.05	1.01±0.24	0.02±0.01	0.02±0.01
H	0.09±0.01	0.02±0.01	0.21±0.08	0.50±0.11	0.63±0.10	0.13±0.03	0.62±0.10	0.08±0.01	0.08±0.01
V	0.07±0.02	0.15±0.05	0.02±0.01	0.56±0.18	0.71±0.20	0.11±0.02	0.70±0.19	0.06±0.01	0.05±0.01
CL	0.06±0.01	0.08±0.02	0.08±0.02	0.12±0.05	0.34±0.12	0.07±0.01	0.35±0.12	0.05±0.01	0.05±0.01
BL	0.15±0.06	0.38±0.16	0.42±0.16	0.56±0.15	0.09±0.04	0.35±0.12	0.07±0.03	0.08±0.02	0.05±0.01

Appendix

Table 59: Navier–Stokes / CNO. All 81 cells: mean

\pm

SD on one line (

n=200

); blue marks matched views. All training views: 500/O500.

	Test observation pattern
Train	R50	H	V	CL	BL	LI	BD	R65	R80
R50	0.03±0.01	0.13±0.04	0.15±0.04	0.53±0.12	0.64±0.14	0.11±0.03	0.61±0.13	0.04±0.01	0.06±0.02
H	0.60±0.05	0.05±0.02	0.78±0.16	0.54±0.11	0.55±0.12	0.46±0.10	0.51±0.10	0.47±0.05	0.31±0.05
V	0.78±0.10	1.25±0.27	0.05±0.02	0.57±0.11	0.59±0.12	0.78±0.19	0.56±0.10	0.66±0.11	0.47±0.10
CL	0.19±0.04	0.71±0.30	0.50±0.14	0.18±0.05	0.40±0.10	0.33±0.11	0.40±0.09	0.18±0.04	0.16±0.04
BL	2.11±0.69	4.18±1.49	3.89±1.46	0.52±0.14	0.09±0.03	3.80±1.40	0.08±0.03	1.47±0.44	0.58±0.16

Appendix

Table 60: Navier–Stokes / DeepONet. All 81 cells: mean

\pm

SD on one line (

n=200

); blue marks matched views. Epoch/cohort in row order: 500/O500, 500/O500, 500/O500, 500/O500, 500/O500, 128/INT, 500/O500, 500/O500, 500/O500.

	Test observation pattern
Train	R50	H	V	CL	BL	LI	BD	R65	R80
R50	0.02±0.01	0.04±0.01	0.04±0.01	0.27±0.08	0.36±0.12	0.03±0.01	0.35±0.12	0.02±0.01	0.02±0.01
H	0.06±0.03	0.07±0.03	0.08±0.03	0.36±0.14	0.47±0.18	0.07±0.03	0.46±0.17	0.06±0.03	0.06±0.03
V	0.03±0.01	0.08±0.04	0.03±0.01	0.34±0.13	0.42±0.13	0.06±0.02	0.39±0.12	0.03±0.01	0.03±0.01
CL	0.11±0.04	0.11±0.04	0.12±0.04	0.19±0.06	0.28±0.10	0.11±0.04	0.35±0.09	0.11±0.04	0.11±0.04
BL	0.09±0.03	0.11±0.04	0.11±0.04	0.46±0.12	0.12±0.05	0.11±0.03	0.11±0.05	0.09±0.03	0.09±0.03

Appendix

Table 61: Navier–Stokes / GNOT. All 81 cells: mean

\pm

SD on one line (

n=200

); blue marks matched views. Epoch/cohort in row order: 200/F200, 200/F200, 200/F200, 200/F200, 200/F200, 200/F200, 200/F200, 500/O500, 200/F200.

	Test observation pattern
Train	R50	H	V	CL	BL	LI	BD	R65	R80
R50	0.03±0.01	0.12±0.06	0.12±0.07	0.62±0.12	0.69±0.10	0.11±0.03	0.69±0.10	0.03±0.01	0.04±0.01
H	0.12±0.02	0.12±0.04	0.28±0.10	0.78±0.23	0.93±0.27	0.17±0.03	0.94±0.27	0.11±0.03	0.10±0.03
V	0.82±0.05	0.90±0.01	0.02±0.01	0.78±0.11	0.83±0.08	0.88±0.02	0.83±0.09	0.63±0.16	0.21±0.21
CL	0.08±0.02	0.13±0.05	0.08±0.01	0.13±0.07	0.27±0.10	0.07±0.01	0.27±0.10	0.07±0.01	0.07±0.01
BL	0.41±0.07	0.34±0.06	0.31±0.05	0.45±0.10	0.06±0.04	0.31±0.05	0.05±0.02	0.47±0.11	0.44±0.08

Appendix

Table 62: Navier–Stokes / Transolver. All 81 cells: mean

\pm

SD on one line (

n=200

); blue marks matched views. Epoch/cohort in row order: 200/M120, 31/P10, 200/M120, 200/M120, 200/M120, 200/M120, 200/M120, 200/M120, 200/M120.

	Test observation pattern
Train	R50	H	V	CL	BL	LI	BD	R65	R80
R50	0.00±0.00	0.05±0.02	0.05±0.02	0.55±0.10	0.64±0.10	0.04±0.01	0.64±0.10	0.03±0.02	0.03±0.02
H	0.33±0.19	0.00±0.00	0.81±0.11	0.54±0.11	0.69±0.11	0.73±0.14	0.68±0.11	0.34±0.19	0.35±0.16
V	1.32±0.62	2.12±0.51	0.00±0.00	2.05±0.52	2.12±0.52	2.02±0.51	2.12±0.49	1.34±0.59	1.33±0.57
CL	0.04±0.01	0.07±0.06	0.06±0.02	0.05±0.02	0.06±0.03	0.06±0.02	0.06±0.03	0.04±0.02	0.04±0.03
BL	0.22±0.12	0.21±0.09	0.18±0.07	0.29±0.09	0.02±0.01	0.12±0.04	0.02±0.01	0.18±0.11	0.13±0.10

Appendix

Table 63: Navier–Stokes / PINO. All 81 cells: mean

\pm

SD on one line (

n=200

); blue marks matched views. All training views: 500/R500.

		Error (%): mean $\pm$ view SD
Pair	Destinations $W$	MIX	Specialist	geo. $R$
Helmholtz / FNO	All nine	$5.31\pm 5.81$	$2.12\pm 1.74$	2.07
Helmholtz / DeepONet	All nine	$12.14\pm 10.68$	$5.01\pm 4.82$	2.56
Heat / DeepONet	All nine	$13.29\pm 6.28$	$9.83\pm 3.78$	1.32
Navier–Stokes / U-FNO	All nine	$3.49\pm 3.98$	$2.55\pm 2.87$	1.19
Poisson / U-FNO	All nine (descriptive)	$7.12\pm 8.01$	$11.65\pm 9.77$	0.46

Appendix

Table 64: Mixed-pattern comparisons with explicit populations. All columns in a row use the same destination set

W

. MIX and specialist errors are mean

\pm

sample SD across the

|W|

cell means, each based on 200 records, multiplied by 100 (error %, not observation density). The geometric mean of

R_{\mathrm{MIX}\to w}

is dimensionless. Poisson’s all-nine and recipe-matched populations are separate; the full-grid

C

remains in Appendix F and is not reused for the restricted set.

	Test observation protocol
Pair	Random 20%	Random 35%	Line 30 / random 20
Helmholtz / FNO	$14.22\pm 3.98$	$4.34\pm 1.44$	$11.50\pm 4.56$
Helmholtz / DeepONet	$18.31\pm 4.18$	$6.73\pm 1.97$	$15.03\pm 5.82$
Heat / DeepONet	$23.21\pm 9.93$	$13.22\pm 5.92$	$18.92\pm 8.73$
Navier–Stokes / U-FNO	$8.39\pm 2.23$	$1.74\pm 0.56$	$6.27\pm 2.80$
Poisson / U-FNO	$15.30\pm 1.80$	$4.47\pm 0.94$	$12.49\pm 7.14$

Appendix

Table 65: MIX on protocols absent from training. Error (%) is

100\times

per-record relative-

L^{2}

, shown as mean

\pm

sample SD over 200 records (

\mathrm{ddof}=1

). Column percentages instead specify observation density. “Line 30 / random 20” assigns one of the two patterns per record. There is no matched specialist for these protocols.

Figure 5: Density response of five completed MIX models. Mean relative-

L^{2}

error at random observation densities of 20%, 35%, 50% (R50), 65% (R65) and 80% (R80); shaded densities were absent from training. Open markers show destination-trained references where available. Masks are separately sampled, not nested; connecting lines do not establish a continuous density law.

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